Properties

Label 336.6.q.j.289.3
Level $336$
Weight $6$
Character 336.289
Analytic conductor $53.889$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,6,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.3
Root \(5.09061 + 8.81720i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.6.q.j.193.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\) \(q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(208.355 + 360.881i) q^{11} +797.918 q^{13} -198.644 q^{15} +(687.775 + 1191.26i) q^{17} +(1156.51 - 2003.14i) q^{19} +(-778.343 - 869.224i) q^{21} +(-477.701 + 827.402i) q^{23} +(1318.92 + 2284.44i) q^{25} -729.000 q^{27} -7035.29 q^{29} +(630.596 + 1092.22i) q^{31} +(-1875.19 + 3247.93i) q^{33} +(891.689 - 2718.92i) q^{35} +(-4888.22 + 8466.65i) q^{37} +(3590.63 + 6219.16i) q^{39} -5400.95 q^{41} -19686.6 q^{43} +(-893.900 - 1548.28i) q^{45} +(1028.28 - 1781.04i) q^{47} +(15391.3 - 6751.63i) q^{49} +(-6189.98 + 10721.4i) q^{51} +(-9011.37 - 15608.2i) q^{53} -9197.45 q^{55} +20817.2 q^{57} +(3717.84 + 6439.49i) q^{59} +(-1747.69 + 3027.09i) q^{61} +(3272.39 - 9978.09i) q^{63} +(-8805.67 + 15251.9i) q^{65} +(7928.21 + 13732.1i) q^{67} -8598.62 q^{69} -58133.5 q^{71} +(-19555.3 - 33870.8i) q^{73} +(-11870.3 + 20560.0i) q^{75} +(-36038.1 - 40246.0i) q^{77} +(4880.35 - 8453.01i) q^{79} +(-3280.50 - 5681.99i) q^{81} +70395.7 q^{83} -30360.6 q^{85} +(-31658.8 - 54834.7i) q^{87} +(-72153.1 + 124973. i) q^{89} +(-101242. + 21229.3i) q^{91} +(-5675.36 + 9830.01i) q^{93} +(25526.1 + 44212.5i) q^{95} -79328.7 q^{97} -33753.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.50000 + 7.79423i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −11.0358 + 19.1146i −0.197414 + 0.341932i −0.947689 0.319194i \(-0.896588\pi\)
0.750275 + 0.661126i \(0.229921\pi\)
\(6\) 0 0
\(7\) −126.882 + 26.6059i −0.978715 + 0.205226i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 208.355 + 360.881i 0.519184 + 0.899254i 0.999751 + 0.0222959i \(0.00709758\pi\)
−0.480567 + 0.876958i \(0.659569\pi\)
\(12\) 0 0
\(13\) 797.918 1.30948 0.654742 0.755853i \(-0.272777\pi\)
0.654742 + 0.755853i \(0.272777\pi\)
\(14\) 0 0
\(15\) −198.644 −0.227955
\(16\) 0 0
\(17\) 687.775 + 1191.26i 0.577197 + 0.999735i 0.995799 + 0.0915652i \(0.0291870\pi\)
−0.418602 + 0.908170i \(0.637480\pi\)
\(18\) 0 0
\(19\) 1156.51 2003.14i 0.734965 1.27300i −0.219774 0.975551i \(-0.570532\pi\)
0.954739 0.297446i \(-0.0961347\pi\)
\(20\) 0 0
\(21\) −778.343 869.224i −0.385144 0.430114i
\(22\) 0 0
\(23\) −477.701 + 827.402i −0.188294 + 0.326135i −0.944682 0.327989i \(-0.893629\pi\)
0.756388 + 0.654124i \(0.226962\pi\)
\(24\) 0 0
\(25\) 1318.92 + 2284.44i 0.422055 + 0.731021i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −7035.29 −1.55341 −0.776707 0.629862i \(-0.783111\pi\)
−0.776707 + 0.629862i \(0.783111\pi\)
\(30\) 0 0
\(31\) 630.596 + 1092.22i 0.117855 + 0.204130i 0.918917 0.394450i \(-0.129065\pi\)
−0.801063 + 0.598581i \(0.795732\pi\)
\(32\) 0 0
\(33\) −1875.19 + 3247.93i −0.299751 + 0.519184i
\(34\) 0 0
\(35\) 891.689 2718.92i 0.123039 0.375168i
\(36\) 0 0
\(37\) −4888.22 + 8466.65i −0.587012 + 1.01673i 0.407610 + 0.913156i \(0.366362\pi\)
−0.994621 + 0.103578i \(0.966971\pi\)
\(38\) 0 0
\(39\) 3590.63 + 6219.16i 0.378015 + 0.654742i
\(40\) 0 0
\(41\) −5400.95 −0.501777 −0.250888 0.968016i \(-0.580723\pi\)
−0.250888 + 0.968016i \(0.580723\pi\)
\(42\) 0 0
\(43\) −19686.6 −1.62367 −0.811837 0.583885i \(-0.801532\pi\)
−0.811837 + 0.583885i \(0.801532\pi\)
\(44\) 0 0
\(45\) −893.900 1548.28i −0.0658048 0.113977i
\(46\) 0 0
\(47\) 1028.28 1781.04i 0.0678997 0.117606i −0.830077 0.557649i \(-0.811704\pi\)
0.897977 + 0.440043i \(0.145037\pi\)
\(48\) 0 0
\(49\) 15391.3 6751.63i 0.915765 0.401715i
\(50\) 0 0
\(51\) −6189.98 + 10721.4i −0.333245 + 0.577197i
\(52\) 0 0
\(53\) −9011.37 15608.2i −0.440658 0.763241i 0.557081 0.830458i \(-0.311921\pi\)
−0.997738 + 0.0672170i \(0.978588\pi\)
\(54\) 0 0
\(55\) −9197.45 −0.409978
\(56\) 0 0
\(57\) 20817.2 0.848664
\(58\) 0 0
\(59\) 3717.84 + 6439.49i 0.139047 + 0.240836i 0.927136 0.374725i \(-0.122263\pi\)
−0.788089 + 0.615561i \(0.788930\pi\)
\(60\) 0 0
\(61\) −1747.69 + 3027.09i −0.0601368 + 0.104160i −0.894526 0.447015i \(-0.852487\pi\)
0.834390 + 0.551175i \(0.185820\pi\)
\(62\) 0 0
\(63\) 3272.39 9978.09i 0.103875 0.316735i
\(64\) 0 0
\(65\) −8805.67 + 15251.9i −0.258511 + 0.447754i
\(66\) 0 0
\(67\) 7928.21 + 13732.1i 0.215769 + 0.373722i 0.953510 0.301361i \(-0.0974410\pi\)
−0.737741 + 0.675083i \(0.764108\pi\)
\(68\) 0 0
\(69\) −8598.62 −0.217423
\(70\) 0 0
\(71\) −58133.5 −1.36861 −0.684306 0.729195i \(-0.739895\pi\)
−0.684306 + 0.729195i \(0.739895\pi\)
\(72\) 0 0
\(73\) −19555.3 33870.8i −0.429495 0.743907i 0.567334 0.823488i \(-0.307975\pi\)
−0.996828 + 0.0795812i \(0.974642\pi\)
\(74\) 0 0
\(75\) −11870.3 + 20560.0i −0.243674 + 0.422055i
\(76\) 0 0
\(77\) −36038.1 40246.0i −0.692684 0.773563i
\(78\) 0 0
\(79\) 4880.35 8453.01i 0.0879798 0.152385i −0.818677 0.574254i \(-0.805292\pi\)
0.906657 + 0.421868i \(0.138626\pi\)
\(80\) 0 0
\(81\) −3280.50 5681.99i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 70395.7 1.12163 0.560816 0.827940i \(-0.310487\pi\)
0.560816 + 0.827940i \(0.310487\pi\)
\(84\) 0 0
\(85\) −30360.6 −0.455788
\(86\) 0 0
\(87\) −31658.8 54834.7i −0.448432 0.776707i
\(88\) 0 0
\(89\) −72153.1 + 124973.i −0.965562 + 1.67240i −0.257464 + 0.966288i \(0.582887\pi\)
−0.708098 + 0.706115i \(0.750446\pi\)
\(90\) 0 0
\(91\) −101242. + 21229.3i −1.28161 + 0.268740i
\(92\) 0 0
\(93\) −5675.36 + 9830.01i −0.0680434 + 0.117855i
\(94\) 0 0
\(95\) 25526.1 + 44212.5i 0.290185 + 0.502616i
\(96\) 0 0
\(97\) −79328.7 −0.856053 −0.428027 0.903766i \(-0.640791\pi\)
−0.428027 + 0.903766i \(0.640791\pi\)
\(98\) 0 0
\(99\) −33753.5 −0.346123
\(100\) 0 0
\(101\) 42416.9 + 73468.2i 0.413747 + 0.716631i 0.995296 0.0968802i \(-0.0308864\pi\)
−0.581549 + 0.813512i \(0.697553\pi\)
\(102\) 0 0
\(103\) 10166.1 17608.3i 0.0944197 0.163540i −0.814947 0.579536i \(-0.803234\pi\)
0.909366 + 0.415996i \(0.136567\pi\)
\(104\) 0 0
\(105\) 25204.5 5285.11i 0.223102 0.0467822i
\(106\) 0 0
\(107\) 3481.09 6029.43i 0.0293938 0.0509116i −0.850954 0.525240i \(-0.823976\pi\)
0.880348 + 0.474328i \(0.157309\pi\)
\(108\) 0 0
\(109\) −56325.7 97559.0i −0.454088 0.786504i 0.544547 0.838730i \(-0.316702\pi\)
−0.998635 + 0.0522262i \(0.983368\pi\)
\(110\) 0 0
\(111\) −87988.0 −0.677823
\(112\) 0 0
\(113\) 112005. 0.825167 0.412583 0.910920i \(-0.364627\pi\)
0.412583 + 0.910920i \(0.364627\pi\)
\(114\) 0 0
\(115\) −10543.6 18262.1i −0.0743439 0.128767i
\(116\) 0 0
\(117\) −32315.7 + 55972.4i −0.218247 + 0.378015i
\(118\) 0 0
\(119\) −118961. 132851.i −0.770083 0.860000i
\(120\) 0 0
\(121\) −6297.91 + 10908.3i −0.0391051 + 0.0677320i
\(122\) 0 0
\(123\) −24304.3 42096.2i −0.144850 0.250888i
\(124\) 0 0
\(125\) −127195. −0.728108
\(126\) 0 0
\(127\) −82224.5 −0.452368 −0.226184 0.974085i \(-0.572625\pi\)
−0.226184 + 0.974085i \(0.572625\pi\)
\(128\) 0 0
\(129\) −88589.5 153442.i −0.468714 0.811837i
\(130\) 0 0
\(131\) −87905.9 + 152258.i −0.447548 + 0.775176i −0.998226 0.0595417i \(-0.981036\pi\)
0.550677 + 0.834718i \(0.314369\pi\)
\(132\) 0 0
\(133\) −93445.8 + 284933.i −0.458069 + 1.39673i
\(134\) 0 0
\(135\) 8045.10 13934.5i 0.0379924 0.0658048i
\(136\) 0 0
\(137\) −15665.2 27132.9i −0.0713072 0.123508i 0.828167 0.560481i \(-0.189384\pi\)
−0.899474 + 0.436973i \(0.856050\pi\)
\(138\) 0 0
\(139\) −152234. −0.668305 −0.334152 0.942519i \(-0.608450\pi\)
−0.334152 + 0.942519i \(0.608450\pi\)
\(140\) 0 0
\(141\) 18509.1 0.0784038
\(142\) 0 0
\(143\) 166250. + 287953.i 0.679863 + 1.17756i
\(144\) 0 0
\(145\) 77640.1 134477.i 0.306666 0.531162i
\(146\) 0 0
\(147\) 121884. + 89580.6i 0.465216 + 0.341917i
\(148\) 0 0
\(149\) −181430. + 314246.i −0.669489 + 1.15959i 0.308558 + 0.951206i \(0.400154\pi\)
−0.978047 + 0.208384i \(0.933180\pi\)
\(150\) 0 0
\(151\) 102563. + 177644.i 0.366056 + 0.634027i 0.988945 0.148283i \(-0.0473746\pi\)
−0.622889 + 0.782310i \(0.714041\pi\)
\(152\) 0 0
\(153\) −111420. −0.384798
\(154\) 0 0
\(155\) −27836.5 −0.0930648
\(156\) 0 0
\(157\) 38636.0 + 66919.5i 0.125096 + 0.216672i 0.921770 0.387736i \(-0.126743\pi\)
−0.796675 + 0.604409i \(0.793409\pi\)
\(158\) 0 0
\(159\) 81102.3 140473.i 0.254414 0.440658i
\(160\) 0 0
\(161\) 38598.1 117692.i 0.117355 0.357836i
\(162\) 0 0
\(163\) 92465.7 160155.i 0.272591 0.472141i −0.696934 0.717136i \(-0.745453\pi\)
0.969525 + 0.244994i \(0.0787861\pi\)
\(164\) 0 0
\(165\) −41388.5 71687.0i −0.118350 0.204989i
\(166\) 0 0
\(167\) −129262. −0.358657 −0.179329 0.983789i \(-0.557393\pi\)
−0.179329 + 0.983789i \(0.557393\pi\)
\(168\) 0 0
\(169\) 265380. 0.714746
\(170\) 0 0
\(171\) 93677.6 + 162254.i 0.244988 + 0.424332i
\(172\) 0 0
\(173\) 253934. 439826.i 0.645067 1.11729i −0.339219 0.940707i \(-0.610163\pi\)
0.984286 0.176582i \(-0.0565040\pi\)
\(174\) 0 0
\(175\) −228127. 254764.i −0.563096 0.628844i
\(176\) 0 0
\(177\) −33460.6 + 57955.4i −0.0802786 + 0.139047i
\(178\) 0 0
\(179\) −66294.5 114825.i −0.154648 0.267859i 0.778283 0.627914i \(-0.216091\pi\)
−0.932931 + 0.360056i \(0.882758\pi\)
\(180\) 0 0
\(181\) −740060. −1.67908 −0.839538 0.543301i \(-0.817174\pi\)
−0.839538 + 0.543301i \(0.817174\pi\)
\(182\) 0 0
\(183\) −31458.5 −0.0694400
\(184\) 0 0
\(185\) −107891. 186873.i −0.231769 0.401436i
\(186\) 0 0
\(187\) −286603. + 496410.i −0.599344 + 1.03809i
\(188\) 0 0
\(189\) 92497.2 19395.7i 0.188354 0.0394958i
\(190\) 0 0
\(191\) −291366. + 504661.i −0.577904 + 1.00096i 0.417815 + 0.908532i \(0.362796\pi\)
−0.995719 + 0.0924273i \(0.970537\pi\)
\(192\) 0 0
\(193\) 200452. + 347193.i 0.387362 + 0.670931i 0.992094 0.125498i \(-0.0400529\pi\)
−0.604731 + 0.796429i \(0.706720\pi\)
\(194\) 0 0
\(195\) −158502. −0.298503
\(196\) 0 0
\(197\) 671589. 1.23293 0.616464 0.787383i \(-0.288564\pi\)
0.616464 + 0.787383i \(0.288564\pi\)
\(198\) 0 0
\(199\) −227511. 394060.i −0.407258 0.705391i 0.587324 0.809352i \(-0.300182\pi\)
−0.994581 + 0.103961i \(0.966848\pi\)
\(200\) 0 0
\(201\) −71353.9 + 123589.i −0.124574 + 0.215769i
\(202\) 0 0
\(203\) 892654. 187180.i 1.52035 0.318801i
\(204\) 0 0
\(205\) 59603.8 103237.i 0.0990580 0.171573i
\(206\) 0 0
\(207\) −38693.8 67019.6i −0.0627647 0.108712i
\(208\) 0 0
\(209\) 963860. 1.52633
\(210\) 0 0
\(211\) 1.19545e6 1.84852 0.924260 0.381764i \(-0.124683\pi\)
0.924260 + 0.381764i \(0.124683\pi\)
\(212\) 0 0
\(213\) −261601. 453105.i −0.395084 0.684306i
\(214\) 0 0
\(215\) 217257. 376300.i 0.320537 0.555186i
\(216\) 0 0
\(217\) −109071. 121806.i −0.157239 0.175598i
\(218\) 0 0
\(219\) 175998. 304837.i 0.247969 0.429495i
\(220\) 0 0
\(221\) 548788. + 950529.i 0.755830 + 1.30914i
\(222\) 0 0
\(223\) −296529. −0.399305 −0.199653 0.979867i \(-0.563981\pi\)
−0.199653 + 0.979867i \(0.563981\pi\)
\(224\) 0 0
\(225\) −213665. −0.281370
\(226\) 0 0
\(227\) 109073. + 188920.i 0.140492 + 0.243340i 0.927682 0.373371i \(-0.121798\pi\)
−0.787190 + 0.616711i \(0.788465\pi\)
\(228\) 0 0
\(229\) 614602. 1.06452e6i 0.774471 1.34142i −0.160620 0.987016i \(-0.551349\pi\)
0.935091 0.354407i \(-0.115317\pi\)
\(230\) 0 0
\(231\) 151515. 461996.i 0.186821 0.569650i
\(232\) 0 0
\(233\) 31472.1 54511.3i 0.0379784 0.0657805i −0.846411 0.532530i \(-0.821242\pi\)
0.884390 + 0.466749i \(0.154575\pi\)
\(234\) 0 0
\(235\) 22695.8 + 39310.4i 0.0268088 + 0.0464341i
\(236\) 0 0
\(237\) 87846.2 0.101590
\(238\) 0 0
\(239\) −219330. −0.248372 −0.124186 0.992259i \(-0.539632\pi\)
−0.124186 + 0.992259i \(0.539632\pi\)
\(240\) 0 0
\(241\) −216932. 375737.i −0.240592 0.416717i 0.720291 0.693672i \(-0.244008\pi\)
−0.960883 + 0.276955i \(0.910675\pi\)
\(242\) 0 0
\(243\) 29524.5 51137.9i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) −40800.4 + 368707.i −0.0434259 + 0.392433i
\(246\) 0 0
\(247\) 922803. 1.59834e6i 0.962424 1.66697i
\(248\) 0 0
\(249\) 316780. + 548680.i 0.323787 + 0.560816i
\(250\) 0 0
\(251\) 1.71109e6 1.71431 0.857155 0.515059i \(-0.172230\pi\)
0.857155 + 0.515059i \(0.172230\pi\)
\(252\) 0 0
\(253\) −398125. −0.391037
\(254\) 0 0
\(255\) −136623. 236638.i −0.131575 0.227894i
\(256\) 0 0
\(257\) −492029. + 852219.i −0.464684 + 0.804856i −0.999187 0.0403103i \(-0.987165\pi\)
0.534503 + 0.845166i \(0.320499\pi\)
\(258\) 0 0
\(259\) 394967. 1.20432e6i 0.365857 1.11556i
\(260\) 0 0
\(261\) 284929. 493512.i 0.258902 0.448432i
\(262\) 0 0
\(263\) 273548. + 473799.i 0.243862 + 0.422381i 0.961811 0.273714i \(-0.0882523\pi\)
−0.717949 + 0.696096i \(0.754919\pi\)
\(264\) 0 0
\(265\) 397791. 0.347969
\(266\) 0 0
\(267\) −1.29876e6 −1.11493
\(268\) 0 0
\(269\) 320556. + 555219.i 0.270099 + 0.467825i 0.968887 0.247504i \(-0.0796102\pi\)
−0.698788 + 0.715329i \(0.746277\pi\)
\(270\) 0 0
\(271\) −274006. + 474593.i −0.226640 + 0.392552i −0.956810 0.290713i \(-0.906107\pi\)
0.730170 + 0.683265i \(0.239441\pi\)
\(272\) 0 0
\(273\) −621054. 693569.i −0.504339 0.563227i
\(274\) 0 0
\(275\) −549607. + 951948.i −0.438249 + 0.759069i
\(276\) 0 0
\(277\) −837070. 1.44985e6i −0.655485 1.13533i −0.981772 0.190062i \(-0.939131\pi\)
0.326287 0.945271i \(-0.394202\pi\)
\(278\) 0 0
\(279\) −102157. −0.0785698
\(280\) 0 0
\(281\) 1.81078e6 1.36804 0.684021 0.729462i \(-0.260230\pi\)
0.684021 + 0.729462i \(0.260230\pi\)
\(282\) 0 0
\(283\) 1.25657e6 + 2.17645e6i 0.932657 + 1.61541i 0.778759 + 0.627323i \(0.215849\pi\)
0.153898 + 0.988087i \(0.450817\pi\)
\(284\) 0 0
\(285\) −229735. + 397913.i −0.167539 + 0.290185i
\(286\) 0 0
\(287\) 685285. 143697.i 0.491096 0.102978i
\(288\) 0 0
\(289\) −236142. + 409009.i −0.166314 + 0.288064i
\(290\) 0 0
\(291\) −356979. 618306.i −0.247121 0.428027i
\(292\) 0 0
\(293\) 107228. 0.0729691 0.0364845 0.999334i \(-0.488384\pi\)
0.0364845 + 0.999334i \(0.488384\pi\)
\(294\) 0 0
\(295\) −164117. −0.109799
\(296\) 0 0
\(297\) −151891. 263082.i −0.0999171 0.173061i
\(298\) 0 0
\(299\) −381166. + 660199.i −0.246568 + 0.427068i
\(300\) 0 0
\(301\) 2.49788e6 523778.i 1.58911 0.333220i
\(302\) 0 0
\(303\) −381752. + 661214.i −0.238877 + 0.413747i
\(304\) 0 0
\(305\) −38574.4 66812.8i −0.0237437 0.0411254i
\(306\) 0 0
\(307\) −1.49622e6 −0.906042 −0.453021 0.891500i \(-0.649654\pi\)
−0.453021 + 0.891500i \(0.649654\pi\)
\(308\) 0 0
\(309\) 182990. 0.109027
\(310\) 0 0
\(311\) 430601. + 745823.i 0.252449 + 0.437255i 0.964200 0.265178i \(-0.0854306\pi\)
−0.711750 + 0.702433i \(0.752097\pi\)
\(312\) 0 0
\(313\) 251969. 436422.i 0.145374 0.251794i −0.784139 0.620586i \(-0.786895\pi\)
0.929512 + 0.368791i \(0.120228\pi\)
\(314\) 0 0
\(315\) 154613. + 172666.i 0.0877952 + 0.0980464i
\(316\) 0 0
\(317\) 240004. 415700.i 0.134144 0.232344i −0.791126 0.611653i \(-0.790505\pi\)
0.925270 + 0.379309i \(0.123838\pi\)
\(318\) 0 0
\(319\) −1.46584e6 2.53890e6i −0.806508 1.39691i
\(320\) 0 0
\(321\) 62659.7 0.0339411
\(322\) 0 0
\(323\) 3.18168e6 1.69688
\(324\) 0 0
\(325\) 1.05239e6 + 1.82280e6i 0.552674 + 0.957259i
\(326\) 0 0
\(327\) 506931. 878031.i 0.262168 0.454088i
\(328\) 0 0
\(329\) −83084.8 + 253341.i −0.0423187 + 0.129037i
\(330\) 0 0
\(331\) −1.09961e6 + 1.90458e6i −0.551656 + 0.955497i 0.446499 + 0.894784i \(0.352671\pi\)
−0.998155 + 0.0607125i \(0.980663\pi\)
\(332\) 0 0
\(333\) −395946. 685799.i −0.195671 0.338911i
\(334\) 0 0
\(335\) −349977. −0.170383
\(336\) 0 0
\(337\) −1.35725e6 −0.651008 −0.325504 0.945541i \(-0.605534\pi\)
−0.325504 + 0.945541i \(0.605534\pi\)
\(338\) 0 0
\(339\) 504023. + 872993.i 0.238205 + 0.412583i
\(340\) 0 0
\(341\) −262775. + 455140.i −0.122377 + 0.211963i
\(342\) 0 0
\(343\) −1.77325e6 + 1.26616e6i −0.813830 + 0.581103i
\(344\) 0 0
\(345\) 94892.6 164359.i 0.0429225 0.0743439i
\(346\) 0 0
\(347\) 1.89970e6 + 3.29038e6i 0.846959 + 1.46698i 0.883909 + 0.467658i \(0.154902\pi\)
−0.0369507 + 0.999317i \(0.511764\pi\)
\(348\) 0 0
\(349\) −1.31753e6 −0.579024 −0.289512 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(350\) 0 0
\(351\) −581682. −0.252010
\(352\) 0 0
\(353\) 1.71253e6 + 2.96618e6i 0.731477 + 1.26695i 0.956252 + 0.292544i \(0.0945018\pi\)
−0.224775 + 0.974411i \(0.572165\pi\)
\(354\) 0 0
\(355\) 641549. 1.11120e6i 0.270184 0.467972i
\(356\) 0 0
\(357\) 500148. 1.52504e6i 0.207696 0.633302i
\(358\) 0 0
\(359\) 735419. 1.27378e6i 0.301161 0.521626i −0.675238 0.737600i \(-0.735959\pi\)
0.976399 + 0.215974i \(0.0692925\pi\)
\(360\) 0 0
\(361\) −1.43700e6 2.48895e6i −0.580347 1.00519i
\(362\) 0 0
\(363\) −113362. −0.0451547
\(364\) 0 0
\(365\) 863235. 0.339154
\(366\) 0 0
\(367\) −2.49039e6 4.31349e6i −0.965168 1.67172i −0.709163 0.705045i \(-0.750927\pi\)
−0.256006 0.966675i \(-0.582407\pi\)
\(368\) 0 0
\(369\) 218738. 378866.i 0.0836295 0.144850i
\(370\) 0 0
\(371\) 1.55865e6 + 1.74064e6i 0.587915 + 0.656561i
\(372\) 0 0
\(373\) −1.98024e6 + 3.42987e6i −0.736962 + 1.27646i 0.216895 + 0.976195i \(0.430407\pi\)
−0.953857 + 0.300261i \(0.902926\pi\)
\(374\) 0 0
\(375\) −572379. 991389.i −0.210187 0.364054i
\(376\) 0 0
\(377\) −5.61359e6 −2.03417
\(378\) 0 0
\(379\) 1.75155e6 0.626359 0.313179 0.949694i \(-0.398606\pi\)
0.313179 + 0.949694i \(0.398606\pi\)
\(380\) 0 0
\(381\) −370010. 640876.i −0.130587 0.226184i
\(382\) 0 0
\(383\) 1.56917e6 2.71788e6i 0.546604 0.946745i −0.451900 0.892068i \(-0.649254\pi\)
0.998504 0.0546771i \(-0.0174129\pi\)
\(384\) 0 0
\(385\) 1.16699e6 244706.i 0.401252 0.0841381i
\(386\) 0 0
\(387\) 797306. 1.38097e6i 0.270612 0.468714i
\(388\) 0 0
\(389\) 526262. + 911512.i 0.176331 + 0.305414i 0.940621 0.339459i \(-0.110244\pi\)
−0.764290 + 0.644872i \(0.776911\pi\)
\(390\) 0 0
\(391\) −1.31420e6 −0.434731
\(392\) 0 0
\(393\) −1.58231e6 −0.516784
\(394\) 0 0
\(395\) 107717. + 186571.i 0.0347370 + 0.0601662i
\(396\) 0 0
\(397\) −227362. + 393803.i −0.0724005 + 0.125401i −0.899953 0.435987i \(-0.856399\pi\)
0.827552 + 0.561389i \(0.189733\pi\)
\(398\) 0 0
\(399\) −2.64134e6 + 553861.i −0.830600 + 0.174168i
\(400\) 0 0
\(401\) 1.44216e6 2.49789e6i 0.447870 0.775733i −0.550378 0.834916i \(-0.685516\pi\)
0.998247 + 0.0591831i \(0.0188496\pi\)
\(402\) 0 0
\(403\) 503164. + 871505.i 0.154329 + 0.267305i
\(404\) 0 0
\(405\) 144812. 0.0438699
\(406\) 0 0
\(407\) −4.07394e6 −1.21907
\(408\) 0 0
\(409\) 112957. + 195647.i 0.0333891 + 0.0578317i 0.882237 0.470805i \(-0.156037\pi\)
−0.848848 + 0.528637i \(0.822703\pi\)
\(410\) 0 0
\(411\) 140986. 244196.i 0.0411692 0.0713072i
\(412\) 0 0
\(413\) −643056. 718141.i −0.185513 0.207174i
\(414\) 0 0
\(415\) −776873. + 1.34558e6i −0.221426 + 0.383522i
\(416\) 0 0
\(417\) −685053. 1.18655e6i −0.192923 0.334152i
\(418\) 0 0
\(419\) −4.31027e6 −1.19941 −0.599707 0.800220i \(-0.704716\pi\)
−0.599707 + 0.800220i \(0.704716\pi\)
\(420\) 0 0
\(421\) 1.25088e6 0.343962 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(422\) 0 0
\(423\) 83290.9 + 144264.i 0.0226332 + 0.0392019i
\(424\) 0 0
\(425\) −1.81424e6 + 3.14236e6i −0.487218 + 0.843887i
\(426\) 0 0
\(427\) 141213. 430583.i 0.0374804 0.114285i
\(428\) 0 0
\(429\) −1.49625e6 + 2.59158e6i −0.392519 + 0.679863i
\(430\) 0 0
\(431\) −2.20397e6 3.81738e6i −0.571494 0.989857i −0.996413 0.0846252i \(-0.973031\pi\)
0.424919 0.905231i \(-0.360303\pi\)
\(432\) 0 0
\(433\) −1.60951e6 −0.412549 −0.206274 0.978494i \(-0.566134\pi\)
−0.206274 + 0.978494i \(0.566134\pi\)
\(434\) 0 0
\(435\) 1.39752e6 0.354108
\(436\) 0 0
\(437\) 1.10493e6 + 1.91380e6i 0.276779 + 0.479395i
\(438\) 0 0
\(439\) −2.21226e6 + 3.83175e6i −0.547867 + 0.948933i 0.450554 + 0.892749i \(0.351227\pi\)
−0.998420 + 0.0561836i \(0.982107\pi\)
\(440\) 0 0
\(441\) −149732. + 1.35311e6i −0.0366622 + 0.331311i
\(442\) 0 0
\(443\) −1.80719e6 + 3.13015e6i −0.437517 + 0.757801i −0.997497 0.0707044i \(-0.977475\pi\)
0.559980 + 0.828506i \(0.310809\pi\)
\(444\) 0 0
\(445\) −1.59254e6 2.75835e6i −0.381232 0.660313i
\(446\) 0 0
\(447\) −3.26574e6 −0.773060
\(448\) 0 0
\(449\) −467024. −0.109326 −0.0546630 0.998505i \(-0.517408\pi\)
−0.0546630 + 0.998505i \(0.517408\pi\)
\(450\) 0 0
\(451\) −1.12531e6 1.94910e6i −0.260515 0.451225i
\(452\) 0 0
\(453\) −923065. + 1.59880e6i −0.211342 + 0.366056i
\(454\) 0 0
\(455\) 711495. 2.16947e6i 0.161118 0.491276i
\(456\) 0 0
\(457\) 300626. 520699.i 0.0673342 0.116626i −0.830393 0.557178i \(-0.811884\pi\)
0.897727 + 0.440552i \(0.145217\pi\)
\(458\) 0 0
\(459\) −501388. 868430.i −0.111082 0.192399i
\(460\) 0 0
\(461\) 2.87193e6 0.629392 0.314696 0.949192i \(-0.398097\pi\)
0.314696 + 0.949192i \(0.398097\pi\)
\(462\) 0 0
\(463\) 2.91502e6 0.631959 0.315979 0.948766i \(-0.397667\pi\)
0.315979 + 0.948766i \(0.397667\pi\)
\(464\) 0 0
\(465\) −125264. 216964.i −0.0268655 0.0465324i
\(466\) 0 0
\(467\) −3.59688e6 + 6.22998e6i −0.763192 + 1.32189i 0.178006 + 0.984029i \(0.443035\pi\)
−0.941197 + 0.337857i \(0.890298\pi\)
\(468\) 0 0
\(469\) −1.37130e6 1.53142e6i −0.287873 0.321486i
\(470\) 0 0
\(471\) −347724. + 602275.i −0.0722241 + 0.125096i
\(472\) 0 0
\(473\) −4.10179e6 7.10451e6i −0.842986 1.46009i
\(474\) 0 0
\(475\) 6.10140e6 1.24078
\(476\) 0 0
\(477\) 1.45984e6 0.293772
\(478\) 0 0
\(479\) −1.39824e6 2.42183e6i −0.278448 0.482286i 0.692551 0.721369i \(-0.256487\pi\)
−0.970999 + 0.239083i \(0.923153\pi\)
\(480\) 0 0
\(481\) −3.90040e6 + 6.75569e6i −0.768682 + 1.33140i
\(482\) 0 0
\(483\) 1.09101e6 228774.i 0.212795 0.0446209i
\(484\) 0 0
\(485\) 875456. 1.51633e6i 0.168997 0.292712i
\(486\) 0 0
\(487\) −1.46624e6 2.53960e6i −0.280144 0.485224i 0.691276 0.722591i \(-0.257049\pi\)
−0.971420 + 0.237367i \(0.923716\pi\)
\(488\) 0 0
\(489\) 1.66438e6 0.314761
\(490\) 0 0
\(491\) 3.06121e6 0.573046 0.286523 0.958073i \(-0.407501\pi\)
0.286523 + 0.958073i \(0.407501\pi\)
\(492\) 0 0
\(493\) −4.83870e6 8.38087e6i −0.896626 1.55300i
\(494\) 0 0
\(495\) 372497. 645183.i 0.0683297 0.118350i
\(496\) 0 0
\(497\) 7.37611e6 1.54669e6i 1.33948 0.280875i
\(498\) 0 0
\(499\) −3.27577e6 + 5.67380e6i −0.588928 + 1.02005i 0.405445 + 0.914119i \(0.367116\pi\)
−0.994373 + 0.105934i \(0.966217\pi\)
\(500\) 0 0
\(501\) −581679. 1.00750e6i −0.103535 0.179329i
\(502\) 0 0
\(503\) 1.58524e6 0.279367 0.139684 0.990196i \(-0.455391\pi\)
0.139684 + 0.990196i \(0.455391\pi\)
\(504\) 0 0
\(505\) −1.87242e6 −0.326719
\(506\) 0 0
\(507\) 1.19421e6 + 2.06843e6i 0.206329 + 0.357373i
\(508\) 0 0
\(509\) −3.38991e6 + 5.87149e6i −0.579953 + 1.00451i 0.415530 + 0.909579i \(0.363596\pi\)
−0.995484 + 0.0949297i \(0.969737\pi\)
\(510\) 0 0
\(511\) 3.38239e6 + 3.77732e6i 0.573022 + 0.639929i
\(512\) 0 0
\(513\) −843098. + 1.46029e6i −0.141444 + 0.244988i
\(514\) 0 0
\(515\) 224383. + 388643.i 0.0372796 + 0.0645702i
\(516\) 0 0
\(517\) 856990. 0.141010
\(518\) 0 0
\(519\) 4.57081e6 0.744859
\(520\) 0 0
\(521\) 5.06918e6 + 8.78008e6i 0.818170 + 1.41711i 0.907029 + 0.421067i \(0.138344\pi\)
−0.0888599 + 0.996044i \(0.528322\pi\)
\(522\) 0 0
\(523\) 49998.6 86600.2i 0.00799289 0.0138441i −0.862001 0.506906i \(-0.830789\pi\)
0.869994 + 0.493062i \(0.164122\pi\)
\(524\) 0 0
\(525\) 959116. 2.92452e6i 0.151870 0.463080i
\(526\) 0 0
\(527\) −867416. + 1.50241e6i −0.136051 + 0.235647i
\(528\) 0 0
\(529\) 2.76178e6 + 4.78354e6i 0.429091 + 0.743207i
\(530\) 0 0
\(531\) −602290. −0.0926978
\(532\) 0 0
\(533\) −4.30952e6 −0.657068
\(534\) 0 0
\(535\) 76833.3 + 133079.i 0.0116055 + 0.0201014i
\(536\) 0 0
\(537\) 596651. 1.03343e6i 0.0892862 0.154648i
\(538\) 0 0
\(539\) 5.64338e6 + 4.14768e6i 0.836695 + 0.614941i
\(540\) 0 0
\(541\) 59871.4 103700.i 0.00879481 0.0152331i −0.861595 0.507597i \(-0.830534\pi\)
0.870389 + 0.492364i \(0.163867\pi\)
\(542\) 0 0
\(543\) −3.33027e6 5.76820e6i −0.484708 0.839538i
\(544\) 0 0
\(545\) 2.48640e6 0.358574
\(546\) 0 0
\(547\) −236568. −0.0338056 −0.0169028 0.999857i \(-0.505381\pi\)
−0.0169028 + 0.999857i \(0.505381\pi\)
\(548\) 0 0
\(549\) −141563. 245194.i −0.0200456 0.0347200i
\(550\) 0 0
\(551\) −8.13641e6 + 1.40927e7i −1.14170 + 1.97749i
\(552\) 0 0
\(553\) −394330. + 1.20238e6i −0.0548336 + 0.167198i
\(554\) 0 0
\(555\) 971018. 1.68185e6i 0.133812 0.231769i
\(556\) 0 0
\(557\) 2.41833e6 + 4.18867e6i 0.330277 + 0.572056i 0.982566 0.185914i \(-0.0595247\pi\)
−0.652289 + 0.757970i \(0.726191\pi\)
\(558\) 0 0
\(559\) −1.57083e7 −2.12617
\(560\) 0 0
\(561\) −5.15885e6 −0.692063
\(562\) 0 0
\(563\) −2.74147e6 4.74837e6i −0.364513 0.631355i 0.624185 0.781277i \(-0.285431\pi\)
−0.988698 + 0.149922i \(0.952098\pi\)
\(564\) 0 0
\(565\) −1.23607e6 + 2.14093e6i −0.162900 + 0.282151i
\(566\) 0 0
\(567\) 567412. + 633664.i 0.0741209 + 0.0827754i
\(568\) 0 0
\(569\) 3.48830e6 6.04191e6i 0.451682 0.782336i −0.546809 0.837258i \(-0.684157\pi\)
0.998491 + 0.0549213i \(0.0174908\pi\)
\(570\) 0 0
\(571\) −5.88573e6 1.01944e7i −0.755458 1.30849i −0.945146 0.326647i \(-0.894081\pi\)
0.189689 0.981844i \(-0.439252\pi\)
\(572\) 0 0
\(573\) −5.24459e6 −0.667306
\(574\) 0 0
\(575\) −2.52020e6 −0.317882
\(576\) 0 0
\(577\) −3.38646e6 5.86552e6i −0.423454 0.733444i 0.572820 0.819681i \(-0.305849\pi\)
−0.996275 + 0.0862365i \(0.972516\pi\)
\(578\) 0 0
\(579\) −1.80407e6 + 3.12474e6i −0.223644 + 0.387362i
\(580\) 0 0
\(581\) −8.93197e6 + 1.87294e6i −1.09776 + 0.230188i
\(582\) 0 0
\(583\) 3.75512e6 6.50407e6i 0.457565 0.792526i
\(584\) 0 0
\(585\) −713259. 1.23540e6i −0.0861703 0.149251i
\(586\) 0 0
\(587\) 1.05020e7 1.25799 0.628996 0.777408i \(-0.283466\pi\)
0.628996 + 0.777408i \(0.283466\pi\)
\(588\) 0 0
\(589\) 2.91717e6 0.346476
\(590\) 0 0
\(591\) 3.02215e6 + 5.23452e6i 0.355916 + 0.616464i
\(592\) 0 0
\(593\) 3.79537e6 6.57378e6i 0.443218 0.767676i −0.554708 0.832045i \(-0.687170\pi\)
0.997926 + 0.0643687i \(0.0205034\pi\)
\(594\) 0 0
\(595\) 3.85223e6 807770.i 0.446087 0.0935396i
\(596\) 0 0
\(597\) 2.04760e6 3.54654e6i 0.235130 0.407258i
\(598\) 0 0
\(599\) −6.50992e6 1.12755e7i −0.741325 1.28401i −0.951892 0.306434i \(-0.900864\pi\)
0.210567 0.977580i \(-0.432469\pi\)
\(600\) 0 0
\(601\) 1.41821e7 1.60160 0.800801 0.598931i \(-0.204408\pi\)
0.800801 + 0.598931i \(0.204408\pi\)
\(602\) 0 0
\(603\) −1.28437e6 −0.143846
\(604\) 0 0
\(605\) −139005. 240764.i −0.0154398 0.0267426i
\(606\) 0 0
\(607\) 6.03862e6 1.04592e7i 0.665221 1.15220i −0.314004 0.949422i \(-0.601671\pi\)
0.979225 0.202775i \(-0.0649961\pi\)
\(608\) 0 0
\(609\) 5.47587e6 + 6.11524e6i 0.598287 + 0.668144i
\(610\) 0 0
\(611\) 820485. 1.42112e6i 0.0889135 0.154003i
\(612\) 0 0
\(613\) 1.36265e6 + 2.36018e6i 0.146465 + 0.253684i 0.929918 0.367766i \(-0.119877\pi\)
−0.783454 + 0.621450i \(0.786544\pi\)
\(614\) 0 0
\(615\) 1.07287e6 0.114382
\(616\) 0 0
\(617\) 8.47094e6 0.895816 0.447908 0.894080i \(-0.352169\pi\)
0.447908 + 0.894080i \(0.352169\pi\)
\(618\) 0 0
\(619\) 8.69174e6 + 1.50545e7i 0.911759 + 1.57921i 0.811578 + 0.584243i \(0.198609\pi\)
0.100180 + 0.994969i \(0.468058\pi\)
\(620\) 0 0
\(621\) 348244. 603176.i 0.0362372 0.0627647i
\(622\) 0 0
\(623\) 5.82995e6 1.77765e7i 0.601789 1.83496i
\(624\) 0 0
\(625\) −2.71793e6 + 4.70759e6i −0.278316 + 0.482058i
\(626\) 0 0
\(627\) 4.33737e6 + 7.51255e6i 0.440613 + 0.763165i
\(628\) 0 0
\(629\) −1.34480e7 −1.35529
\(630\) 0 0
\(631\) 6.45149e6 0.645040 0.322520 0.946563i \(-0.395470\pi\)
0.322520 + 0.946563i \(0.395470\pi\)
\(632\) 0 0
\(633\) 5.37951e6 + 9.31759e6i 0.533622 + 0.924260i
\(634\) 0 0
\(635\) 907413. 1.57169e6i 0.0893040 0.154679i
\(636\) 0 0
\(637\) 1.22810e7 5.38725e6i 1.19918 0.526039i
\(638\) 0 0
\(639\) 2.35440e6 4.07795e6i 0.228102 0.395084i
\(640\) 0 0
\(641\) −8.88581e6 1.53907e7i −0.854185 1.47949i −0.877399 0.479761i \(-0.840723\pi\)
0.0232142 0.999731i \(-0.492610\pi\)
\(642\) 0 0
\(643\) 9.34806e6 0.891649 0.445825 0.895120i \(-0.352910\pi\)
0.445825 + 0.895120i \(0.352910\pi\)
\(644\) 0 0
\(645\) 3.91063e6 0.370124
\(646\) 0 0
\(647\) 2.67193e6 + 4.62792e6i 0.250937 + 0.434635i 0.963784 0.266684i \(-0.0859280\pi\)
−0.712847 + 0.701319i \(0.752595\pi\)
\(648\) 0 0
\(649\) −1.54926e6 + 2.68340e6i −0.144382 + 0.250077i
\(650\) 0 0
\(651\) 458567. 1.39825e6i 0.0424083 0.129310i
\(652\) 0 0
\(653\) 598506. 1.03664e6i 0.0549270 0.0951363i −0.837255 0.546813i \(-0.815841\pi\)
0.892182 + 0.451677i \(0.149174\pi\)
\(654\) 0 0
\(655\) −1.94022e6 3.36057e6i −0.176705 0.306062i
\(656\) 0 0
\(657\) 3.16796e6 0.286330
\(658\) 0 0
\(659\) 1.17541e7 1.05433 0.527163 0.849764i \(-0.323256\pi\)
0.527163 + 0.849764i \(0.323256\pi\)
\(660\) 0 0
\(661\) 9.00573e6 + 1.55984e7i 0.801706 + 1.38860i 0.918493 + 0.395438i \(0.129407\pi\)
−0.116787 + 0.993157i \(0.537259\pi\)
\(662\) 0 0
\(663\) −4.93910e6 + 8.55476e6i −0.436379 + 0.755830i
\(664\) 0 0
\(665\) −4.41512e6 4.93064e6i −0.387158 0.432364i
\(666\) 0 0
\(667\) 3.36076e6 5.82102e6i 0.292498 0.506622i
\(668\) 0 0
\(669\) −1.33438e6 2.31122e6i −0.115270 0.199653i
\(670\) 0 0
\(671\) −1.45656e6 −0.124888
\(672\) 0 0
\(673\) 1.40977e7 1.19981 0.599904 0.800072i \(-0.295206\pi\)
0.599904 + 0.800072i \(0.295206\pi\)
\(674\) 0 0
\(675\) −961494. 1.66536e6i −0.0812245 0.140685i
\(676\) 0 0
\(677\) −3.82793e6 + 6.63018e6i −0.320991 + 0.555973i −0.980693 0.195555i \(-0.937349\pi\)
0.659702 + 0.751527i \(0.270683\pi\)
\(678\) 0 0
\(679\) 1.00654e7 2.11061e6i 0.837832 0.175684i
\(680\) 0 0
\(681\) −981657. + 1.70028e6i −0.0811133 + 0.140492i
\(682\) 0 0
\(683\) −5.43378e6 9.41158e6i −0.445708 0.771988i 0.552394 0.833583i \(-0.313715\pi\)
−0.998101 + 0.0615953i \(0.980381\pi\)
\(684\) 0 0
\(685\) 691510. 0.0563083
\(686\) 0 0
\(687\) 1.10628e7 0.894283
\(688\) 0 0
\(689\) −7.19034e6 1.24540e7i −0.577034 0.999452i
\(690\) 0 0
\(691\) −7.30588e6 + 1.26542e7i −0.582073 + 1.00818i 0.413160 + 0.910658i \(0.364425\pi\)
−0.995233 + 0.0975219i \(0.968908\pi\)
\(692\) 0 0
\(693\) 4.28272e6 898040.i 0.338756 0.0710334i
\(694\) 0 0
\(695\) 1.68002e6 2.90989e6i 0.131933 0.228515i
\(696\) 0 0
\(697\) −3.71464e6 6.43395e6i −0.289624 0.501644i
\(698\) 0 0
\(699\) 566498. 0.0438536
\(700\) 0 0
\(701\) 1.90104e7 1.46115 0.730577 0.682830i \(-0.239251\pi\)
0.730577 + 0.682830i \(0.239251\pi\)
\(702\) 0 0
\(703\) 1.13066e7 + 1.95836e7i 0.862866 + 1.49453i
\(704\) 0 0
\(705\) −204263. + 353793.i −0.0154780 + 0.0268088i
\(706\) 0 0
\(707\) −7.33664e6 8.19328e6i −0.552012 0.616466i
\(708\) 0 0
\(709\) 5.79644e6 1.00397e7i 0.433058 0.750078i −0.564077 0.825722i \(-0.690768\pi\)
0.997135 + 0.0756440i \(0.0241013\pi\)
\(710\) 0 0
\(711\) 395308. + 684693.i 0.0293266 + 0.0507951i
\(712\) 0 0
\(713\) −1.20494e6 −0.0887653
\(714\) 0 0
\(715\) −7.33881e6 −0.536859
\(716\) 0 0
\(717\) −986984. 1.70951e6i −0.0716989 0.124186i
\(718\) 0 0
\(719\) −5.14304e6 + 8.90800e6i −0.371020 + 0.642626i −0.989723 0.142999i \(-0.954325\pi\)
0.618703 + 0.785625i \(0.287659\pi\)
\(720\) 0 0
\(721\) −821420. + 2.50466e6i −0.0588474 + 0.179436i
\(722\) 0 0
\(723\) 1.95239e6 3.38163e6i 0.138906 0.240592i
\(724\) 0 0
\(725\) −9.27900e6 1.60717e7i −0.655626 1.13558i
\(726\) 0 0
\(727\) −1.00970e7 −0.708526 −0.354263 0.935146i \(-0.615268\pi\)
−0.354263 + 0.935146i \(0.615268\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.35399e7 2.34518e7i −0.937180 1.62324i
\(732\) 0 0
\(733\) −962010. + 1.66625e6i −0.0661332 + 0.114546i −0.897196 0.441632i \(-0.854400\pi\)
0.831063 + 0.556178i \(0.187733\pi\)
\(734\) 0 0
\(735\) −3.05739e6 + 1.34117e6i −0.208753 + 0.0915728i
\(736\) 0 0
\(737\) −3.30376e6 + 5.72228e6i −0.224047 + 0.388061i
\(738\) 0 0
\(739\) 2.10136e6 + 3.63967e6i 0.141543 + 0.245161i 0.928078 0.372386i \(-0.121460\pi\)
−0.786535 + 0.617546i \(0.788127\pi\)
\(740\) 0 0
\(741\) 1.66104e7 1.11131
\(742\) 0 0
\(743\) 1.99659e7 1.32684 0.663418 0.748249i \(-0.269105\pi\)
0.663418 + 0.748249i \(0.269105\pi\)
\(744\) 0 0
\(745\) −4.00445e6 6.93592e6i −0.264334 0.457839i
\(746\) 0 0
\(747\) −2.85102e6 + 4.93812e6i −0.186939 + 0.323787i
\(748\) 0 0
\(749\) −281271. + 857646.i −0.0183198 + 0.0558603i
\(750\) 0 0
\(751\) −1.81988e6 + 3.15212e6i −0.117745 + 0.203940i −0.918874 0.394552i \(-0.870900\pi\)
0.801129 + 0.598492i \(0.204233\pi\)
\(752\) 0 0
\(753\) 7.69992e6 + 1.33366e7i 0.494878 + 0.857155i
\(754\) 0 0
\(755\) −4.52745e6 −0.289059
\(756\) 0 0
\(757\) 1.73429e7 1.09997 0.549986 0.835174i \(-0.314633\pi\)
0.549986 + 0.835174i \(0.314633\pi\)
\(758\) 0 0
\(759\) −1.79156e6 3.10308e6i −0.112883 0.195519i
\(760\) 0 0
\(761\) 5.17841e6 8.96927e6i 0.324142 0.561430i −0.657196 0.753719i \(-0.728258\pi\)
0.981338 + 0.192289i \(0.0615912\pi\)
\(762\) 0 0
\(763\) 9.74238e6 + 1.08799e7i 0.605834 + 0.676572i
\(764\) 0 0
\(765\) 1.22960e6 2.12974e6i 0.0759647 0.131575i
\(766\) 0 0
\(767\) 2.96653e6 + 5.13818e6i 0.182079 + 0.315371i
\(768\) 0 0
\(769\) −1.81548e7 −1.10707 −0.553534 0.832826i \(-0.686721\pi\)
−0.553534 + 0.832826i \(0.686721\pi\)
\(770\) 0 0
\(771\) −8.85652e6 −0.536571
\(772\) 0 0
\(773\) 8.17830e6 + 1.41652e7i 0.492282 + 0.852658i 0.999960 0.00888861i \(-0.00282937\pi\)
−0.507678 + 0.861547i \(0.669496\pi\)
\(774\) 0 0
\(775\) −1.66341e6 + 2.88112e6i −0.0994823 + 0.172308i
\(776\) 0 0
\(777\) 1.11641e7 2.34100e6i 0.663395 0.139107i
\(778\) 0 0
\(779\) −6.24627e6 + 1.08189e7i −0.368788 + 0.638760i
\(780\) 0 0
\(781\) −1.21124e7 2.09793e7i −0.710562 1.23073i
\(782\) 0 0
\(783\) 5.12873e6 0.298955
\(784\) 0 0
\(785\) −1.70552e6 −0.0987829
\(786\) 0 0
\(787\) −2.22364e6 3.85146e6i −0.127976 0.221661i 0.794916 0.606719i \(-0.207515\pi\)
−0.922892 + 0.385058i \(0.874181\pi\)
\(788\) 0 0
\(789\) −2.46193e6 + 4.26419e6i −0.140794 + 0.243862i
\(790\) 0 0
\(791\) −1.42115e7 + 2.97999e6i −0.807603 + 0.169346i
\(792\) 0 0
\(793\) −1.39452e6 + 2.41537e6i −0.0787481 + 0.136396i
\(794\) 0 0
\(795\) 1.79006e6 + 3.10047e6i 0.100450 + 0.173984i
\(796\) 0 0
\(797\) −9.15303e6 −0.510410 −0.255205 0.966887i \(-0.582143\pi\)
−0.255205 + 0.966887i \(0.582143\pi\)
\(798\) 0 0
\(799\) 2.82891e6 0.156766
\(800\) 0 0
\(801\) −5.84440e6 1.01228e7i −0.321854 0.557467i
\(802\) 0 0
\(803\) 8.14889e6 1.41143e7i 0.445974 0.772450i
\(804\) 0 0
\(805\) 1.82368e6 + 2.03662e6i 0.0991879 + 0.110769i
\(806\) 0 0
\(807\) −2.88500e6 + 4.99697e6i −0.155942 + 0.270099i
\(808\) 0 0
\(809\) −1.25962e7 2.18172e7i −0.676654 1.17200i −0.975982 0.217849i \(-0.930096\pi\)
0.299328 0.954150i \(-0.403237\pi\)
\(810\) 0 0
\(811\) 8.90585e6 0.475470 0.237735 0.971330i \(-0.423595\pi\)
0.237735 + 0.971330i \(0.423595\pi\)
\(812\) 0 0
\(813\) −4.93211e6 −0.261702
\(814\) 0 0
\(815\) 2.04087e6 + 3.53488e6i 0.107627 + 0.186415i
\(816\) 0 0
\(817\) −2.27678e7 + 3.94349e7i −1.19334 + 2.06693i
\(818\) 0 0
\(819\) 2.61110e6 7.96170e6i 0.136023 0.414759i
\(820\) 0 0
\(821\) −1.67745e7 + 2.90543e7i −0.868543 + 1.50436i −0.00505812 + 0.999987i \(0.501610\pi\)
−0.863485 + 0.504374i \(0.831723\pi\)
\(822\) 0 0
\(823\) −7.30006e6 1.26441e7i −0.375688 0.650710i 0.614742 0.788728i \(-0.289260\pi\)
−0.990430 + 0.138018i \(0.955927\pi\)
\(824\) 0 0
\(825\) −9.89293e6 −0.506046
\(826\) 0 0
\(827\) −1.97530e7 −1.00431 −0.502156 0.864777i \(-0.667460\pi\)
−0.502156 + 0.864777i \(0.667460\pi\)
\(828\) 0 0
\(829\) 2.03127e6 + 3.51827e6i 0.102655 + 0.177804i 0.912778 0.408456i \(-0.133933\pi\)
−0.810122 + 0.586261i \(0.800599\pi\)
\(830\) 0 0
\(831\) 7.53363e6 1.30486e7i 0.378444 0.655485i
\(832\) 0 0
\(833\) 1.86287e7 + 1.36914e7i 0.930186 + 0.683653i
\(834\) 0 0
\(835\) 1.42651e6 2.47079e6i 0.0708041 0.122636i
\(836\) 0 0
\(837\) −459704. 796231.i −0.0226811 0.0392849i
\(838\) 0 0
\(839\) 1.60509e7 0.787217 0.393609 0.919278i \(-0.371227\pi\)
0.393609 + 0.919278i \(0.371227\pi\)
\(840\) 0 0
\(841\) 2.89842e7 1.41309
\(842\) 0 0
\(843\) 8.14850e6 + 1.41136e7i 0.394920 + 0.684021i
\(844\) 0 0
\(845\) −2.92868e6 + 5.07263e6i −0.141101 + 0.244394i
\(846\) 0 0
\(847\) 508869. 1.55163e6i 0.0243724 0.0743157i
\(848\) 0 0
\(849\) −1.13092e7 + 1.95881e7i −0.538470 + 0.932657i
\(850\) 0 0
\(851\) −4.67022e6 8.08905e6i −0.221061 0.382890i
\(852\) 0 0
\(853\) −1.23887e7 −0.582979 −0.291489 0.956574i \(-0.594151\pi\)
−0.291489 + 0.956574i \(0.594151\pi\)
\(854\) 0 0
\(855\) −4.13523e6 −0.193457
\(856\) 0 0
\(857\) −1.31760e7 2.28214e7i −0.612816 1.06143i −0.990764 0.135601i \(-0.956703\pi\)
0.377948 0.925827i \(-0.376630\pi\)
\(858\) 0 0
\(859\) −6.06245e6 + 1.05005e7i −0.280327 + 0.485541i −0.971465 0.237182i \(-0.923776\pi\)
0.691138 + 0.722723i \(0.257110\pi\)
\(860\) 0 0
\(861\) 4.20379e6 + 4.69463e6i 0.193256 + 0.215821i
\(862\) 0 0
\(863\) 1.27976e6 2.21661e6i 0.0584927 0.101312i −0.835296 0.549800i \(-0.814704\pi\)
0.893789 + 0.448488i \(0.148037\pi\)
\(864\) 0 0
\(865\) 5.60472e6 + 9.70766e6i 0.254691 + 0.441138i
\(866\) 0 0
\(867\) −4.25055e6 −0.192042
\(868\) 0 0
\(869\) 4.06737e6 0.182711
\(870\) 0 0
\(871\) 6.32606e6 + 1.09571e7i 0.282545 + 0.489383i
\(872\) 0 0
\(873\) 3.21281e6 5.56475e6i 0.142676 0.247121i
\(874\) 0 0
\(875\) 1.61388e7 3.38414e6i 0.712610 0.149427i
\(876\) 0 0
\(877\) −1.32910e7 + 2.30207e7i −0.583523 + 1.01069i 0.411535 + 0.911394i \(0.364993\pi\)
−0.995058 + 0.0992977i \(0.968340\pi\)
\(878\) 0 0
\(879\) 482526. + 835759.i 0.0210644 + 0.0364845i
\(880\) 0 0
\(881\) −8.32262e6 −0.361260 −0.180630 0.983551i \(-0.557814\pi\)
−0.180630 + 0.983551i \(0.557814\pi\)
\(882\) 0 0
\(883\) −1.81133e7 −0.781798 −0.390899 0.920434i \(-0.627836\pi\)
−0.390899 + 0.920434i \(0.627836\pi\)
\(884\) 0 0
\(885\) −738528. 1.27917e6i −0.0316963 0.0548996i
\(886\) 0 0
\(887\) −1.02128e7 + 1.76890e7i −0.435847 + 0.754909i −0.997364 0.0725558i \(-0.976884\pi\)
0.561517 + 0.827465i \(0.310218\pi\)
\(888\) 0 0
\(889\) 1.04328e7 2.18765e6i 0.442739 0.0928377i
\(890\) 0 0
\(891\) 1.36702e6 2.36774e6i 0.0576872 0.0999171i
\(892\) 0 0
\(893\) −2.37844e6 4.11959e6i −0.0998078 0.172872i
\(894\) 0 0
\(895\) 2.92645e6 0.122119
\(896\) 0 0
\(897\) −6.86099e6 −0.284712
\(898\) 0 0
\(899\) −4.43642e6 7.68411e6i −0.183077 0.317099i
\(900\) 0 0
\(901\) 1.23956e7 2.14698e7i 0.508693 0.881082i
\(902\) 0 0
\(903\) 1.53229e7 + 1.71120e7i 0.625347 + 0.698364i
\(904\) 0 0
\(905\) 8.16715e6 1.41459e7i 0.331474 0.574130i
\(906\) 0 0
\(907\) −1.00599e7 1.74242e7i −0.406044 0.703290i 0.588398 0.808571i \(-0.299759\pi\)
−0.994442 + 0.105282i \(0.966426\pi\)
\(908\) 0 0
\(909\) −6.87153e6 −0.275832
\(910\) 0 0
\(911\) −3.17075e7 −1.26580 −0.632902 0.774232i \(-0.718137\pi\)
−0.632902 + 0.774232i \(0.718137\pi\)
\(912\) 0 0
\(913\) 1.46673e7 + 2.54045e7i 0.582334 + 1.00863i
\(914\) 0 0
\(915\) 347169. 601315.i 0.0137085 0.0237437i
\(916\) 0 0
\(917\) 7.10277e6 2.16576e7i 0.278936 0.850525i
\(918\) 0 0
\(919\) −4.35364e6 + 7.54072e6i −0.170045 + 0.294526i −0.938435 0.345455i \(-0.887725\pi\)
0.768390 + 0.639981i \(0.221058\pi\)
\(920\) 0 0
\(921\) −6.73297e6 1.16619e7i −0.261552 0.453021i
\(922\) 0 0
\(923\) −4.63857e7 −1.79217
\(924\) 0 0
\(925\) −2.57887e7 −0.991005
\(926\) 0 0
\(927\) 823457. + 1.42627e6i 0.0314732 + 0.0545133i
\(928\) 0 0
\(929\) −1.62021e7 + 2.80629e7i −0.615931 + 1.06682i 0.374289 + 0.927312i \(0.377887\pi\)
−0.990220 + 0.139512i \(0.955447\pi\)
\(930\) 0 0
\(931\) 4.27574e6 3.86392e7i 0.161673 1.46101i
\(932\) 0 0
\(933\) −3.87541e6 + 6.71241e6i −0.145752 + 0.252449i
\(934\) 0 0
\(935\) −6.32578e6 1.09566e7i −0.236638 0.409869i
\(936\) 0 0
\(937\) −2.19155e7 −0.815458 −0.407729 0.913103i \(-0.633679\pi\)
−0.407729 + 0.913103i \(0.633679\pi\)
\(938\) 0 0
\(939\) 4.53544e6 0.167863
\(940\) 0 0
\(941\) −7.30095e6 1.26456e7i −0.268785 0.465550i 0.699763 0.714375i \(-0.253289\pi\)
−0.968548 + 0.248825i \(0.919956\pi\)
\(942\) 0 0
\(943\) 2.58004e6 4.46876e6i 0.0944815 0.163647i
\(944\) 0 0
\(945\) −650041. + 1.98209e6i −0.0236789 + 0.0722012i
\(946\) 0 0
\(947\) −1.04189e7 + 1.80461e7i −0.377527 + 0.653895i −0.990702 0.136052i \(-0.956559\pi\)
0.613175 + 0.789947i \(0.289892\pi\)
\(948\) 0 0
\(949\) −1.56035e7 2.70261e7i −0.562416 0.974134i
\(950\) 0 0
\(951\) 4.32008e6 0.154896
\(952\) 0 0
\(953\) −942012. −0.0335988 −0.0167994 0.999859i \(-0.505348\pi\)
−0.0167994 + 0.999859i \(0.505348\pi\)
\(954\) 0 0
\(955\) −6.43092e6 1.11387e7i −0.228173 0.395208i
\(956\) 0 0
\(957\) 1.31925e7 2.28501e7i 0.465638 0.806508i
\(958\) 0 0
\(959\) 2.70953e6 + 3.02589e6i 0.0951364 + 0.106245i
\(960\) 0 0
\(961\) 1.35193e7 2.34161e7i 0.472221 0.817910i
\(962\) 0 0
\(963\) 281969. + 488384.i 0.00979795 + 0.0169705i
\(964\) 0 0
\(965\) −8.84860e6 −0.305884
\(966\) 0 0
\(967\) 2.56570e7 0.882346 0.441173 0.897422i \(-0.354562\pi\)
0.441173 + 0.897422i \(0.354562\pi\)
\(968\) 0 0
\(969\) 1.43176e7 + 2.47988e7i 0.489847 + 0.848439i
\(970\) 0 0
\(971\) −1.07137e7 + 1.85566e7i −0.364662 + 0.631613i −0.988722 0.149763i \(-0.952149\pi\)
0.624060 + 0.781377i \(0.285482\pi\)
\(972\) 0 0
\(973\) 1.93158e7 4.05032e6i 0.654080 0.137153i
\(974\) 0 0
\(975\) −9.47153e6 + 1.64052e7i −0.319086 + 0.552674i
\(976\) 0 0
\(977\) 1.02420e7 + 1.77397e7i 0.343281 + 0.594580i 0.985040 0.172326i \(-0.0551283\pi\)
−0.641759 + 0.766906i \(0.721795\pi\)
\(978\) 0 0
\(979\) −6.01338e7 −2.00522
\(980\) 0 0
\(981\) 9.12476e6 0.302726
\(982\) 0 0
\(983\) 1.38825e7 + 2.40452e7i 0.458230 + 0.793677i 0.998868 0.0475784i \(-0.0151504\pi\)
−0.540638 + 0.841255i \(0.681817\pi\)
\(984\) 0 0
\(985\) −7.41152e6 + 1.28371e7i −0.243398 + 0.421578i
\(986\) 0 0
\(987\) −2.34848e6 + 492450.i −0.0767349 + 0.0160905i
\(988\) 0 0
\(989\) 9.40429e6 1.62887e7i 0.305728 0.529536i
\(990\) 0 0
\(991\) 1.73416e7 + 3.00366e7i 0.560926 + 0.971552i 0.997416 + 0.0718429i \(0.0228880\pi\)
−0.436490 + 0.899709i \(0.643779\pi\)
\(992\) 0 0
\(993\) −1.97930e7 −0.636998
\(994\) 0 0
\(995\) 1.00431e7 0.321594
\(996\) 0 0
\(997\) 7.42861e6 + 1.28667e7i 0.236685 + 0.409950i 0.959761 0.280818i \(-0.0906059\pi\)
−0.723076 + 0.690768i \(0.757273\pi\)
\(998\) 0 0
\(999\) 3.56351e6 6.17219e6i 0.112970 0.195671i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 336.6.q.j.289.3 8
4.3 odd 2 21.6.e.c.16.4 yes 8
7.4 even 3 inner 336.6.q.j.193.3 8
12.11 even 2 63.6.e.e.37.1 8
28.3 even 6 147.6.e.o.67.4 8
28.11 odd 6 21.6.e.c.4.4 8
28.19 even 6 147.6.a.l.1.1 4
28.23 odd 6 147.6.a.m.1.1 4
28.27 even 2 147.6.e.o.79.4 8
84.11 even 6 63.6.e.e.46.1 8
84.23 even 6 441.6.a.w.1.4 4
84.47 odd 6 441.6.a.v.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.4 8 28.11 odd 6
21.6.e.c.16.4 yes 8 4.3 odd 2
63.6.e.e.37.1 8 12.11 even 2
63.6.e.e.46.1 8 84.11 even 6
147.6.a.l.1.1 4 28.19 even 6
147.6.a.m.1.1 4 28.23 odd 6
147.6.e.o.67.4 8 28.3 even 6
147.6.e.o.79.4 8 28.27 even 2
336.6.q.j.193.3 8 7.4 even 3 inner
336.6.q.j.289.3 8 1.1 even 1 trivial
441.6.a.v.1.4 4 84.47 odd 6
441.6.a.w.1.4 4 84.23 even 6